Yes, everything and all apps work on GrapheneOS if you decide to install the services and framework. GrapheneOS is not rooted/jailbroken.
But it doesn't have to be Google Authenticator, use a FOSS alternative like Aegis from F-Droid.
Discussion
+1 for Aegis. Way better than google Authenticator. You can directly import all codes from authenticator to Aegis.
Google enabling cloud and internet support for authenticator now is sketchy imo